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Characterizations of error bounds for lower semicontinuous functions on metric spaces

Dominique Azé, Jean-Noël Corvellec (2004)

ESAIM: Control, Optimisation and Calculus of Variations

Refining the variational method introduced in Azé et al. [Nonlinear Anal. 49 (2002) 643-670], we give characterizations of the existence of so-called global and local error bounds, for lower semicontinuous functions defined on complete metric spaces. We thus provide a systematic and synthetic approach to the subject, emphasizing the special case of convex functions defined on arbitrary Banach spaces (refining the abstract part of Azé and Corvellec [SIAM J. Optim. 12 (2002) 913-927], and the characterization...

Characterizations of error bounds for lower semicontinuous functions on metric spaces

Dominique Azé, Jean-Noël Corvellec (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Refining the variational method introduced in Azé et al. [Nonlinear Anal. 49 (2002) 643-670], we give characterizations of the existence of so-called global and local error bounds, for lower semicontinuous functions defined on complete metric spaces. We thus provide a systematic and synthetic approach to the subject, emphasizing the special case of convex functions defined on arbitrary Banach spaces (refining the abstract part of Azé and Corvellec [SIAM J. Optim. 12 (2002) 913-927], and the characterization...

Characterizations of the Solution Sets of Generalized Convex Minimization Problems

Ivanov, Vsevolod (2003)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 90C26, 90C20, 49J52, 47H05, 47J20.In this paper we obtain some simple characterizations of the solution sets of a pseudoconvex program and a variational inequality. Similar characterizations of the solution set of a quasiconvex quadratic program are derived. Applications of these characterizations are given.

Construction of a Φ-function for two convex polytopes

Y. Stoyan, J. Terno, M. Gil, T. Romanova, G. Scheithauer (2002)

Applicationes Mathematicae

The analytical description of Φ-functions for two convex polytopes is investigated. These Φ-functions can be used for mathematical modelling of packing problems in the three-dimensional space. Only translations of the polytopes are considered. The approach consists of two stages. First the 0-level surface of a Φ-function is constructed, and secondly, the surface is extended to get the Φ-function. The method for constructing the 0-level surface is described in detail.

Continuous reformulations and heuristics for the euclidean travelling salesperson problem

Tuomo Valkonen, Tommi Kärkkäinen (2009)

ESAIM: Control, Optimisation and Calculus of Variations

We consider continuous reformulations of the euclidean travelling salesperson problem (TSP), based on certain clustering problem formulations. These reformulations allow us to apply a generalisation with perturbations of the Weiszfeld algorithm in an attempt to find local approximate solutions to the euclidean TSP.

Continuous reformulations and heuristics for the Euclidean travelling salesperson problem

Tuomo Valkonen, Tommi Kärkkäinen (2008)

ESAIM: Control, Optimisation and Calculus of Variations

We consider continuous reformulations of the Euclidean travelling salesperson problem (TSP), based on certain clustering problem formulations. These reformulations allow us to apply a generalisation with perturbations of the Weiszfeld algorithm in an attempt to find local approximate solutions to the Euclidean TSP.

Convex quadratic underestimation and Branch and Bound for univariate global optimization with one nonconvex constraint

Hoai An Le Thi, Mohand Ouanes (2006)

RAIRO - Operations Research

The purpose of this paper is to demonstrate that, for globally minimize one dimensional nonconvex problems with both twice differentiable function and constraint, we can propose an efficient algorithm based on Branch and Bound techniques. The method is first displayed in the simple case with an interval constraint. The extension is displayed afterwards to the general case with an additional nonconvex twice differentiable constraint. A quadratic bounding function which is better than the well known...

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